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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets

Tao Sun
/ Qingguo Li
/ Nianbai Fan
Published Online: 2018-09-18 | DOI: https://doi.org/10.1515/math-2018-0090

## Abstract

In this paper, we first introduce the notion of 𝓜𝓝-convergence in posets as an unified form of O-convergence and O2-convergence. Then, by studying the fundamental properties of 𝓜𝓝-topology which is determined by 𝓜𝓝-convergence according to the standard topological approach, an equivalent characterization to the 𝓜𝓝-convergence being topological is established. Finally, the lim-inf𝓜-convergence in posets is further investigated, and a sufficient and necessary condition for lim-inf𝓜-convergence to be topological is obtained.

MSC 2010: 54A20; 06A06

## 1 Introduction, Notations and Preliminaries

The concept of O-convergence in partially ordered sets (posets, for short) was introduced by Birkhoff [1], Frink [2] and Mcshane [3]. It is defined as follows: a net (xi)iI in a poset P is said to O-converge to xP if there exist subsets D and F of P such that

1. D is directed and F is filtered;

2. sup D = x = inf F;

3. for every dD and eF, dxie holds eventually, i.e., there exists i0I such that dxie for all ii0.

As what has been showed in [4], the O-convergence (Note: in [4], the O-convergence is called order-convergence) in a general poset P may not be topological, i.e., it is possible that P can not be endowed with a topology such that the O-convergence and the associated topological convergence are consistent. Hence, much work has been done to characterize those special posets in which the O-convergence is topological. The most recent result in [5] shows that the O-convergence in a poset which satisfies Condition (△) is topological if and only if the poset is 𝓞-doubly continuous. This means that for a special class of posets, a sufficient and necessary condition for O-convergence being topological is obtained.

As a direct generalization of O-convergence, O2-convergence in posets has been discussed in [11] from the order-theoretical point of view. It is defined as follows: a net (xi)iI in a poset P is said to O2-converge to xP if there exist subsets A and B of P such that

1. sup A = x = inf B;

2. for every aA and bB, axib holds eventually.

In fact, the O2-convergence is also not topological generally. To clarify those special posets in which the O2-convergence is topological, Zhao and Li [6] showed that for any poset P satisfying Condition (∗), O2-convergence is topological if and only if P is α-doubly continuous. As a further result, Li and Zou [7] proved that the O2-convergence in a poset P is topological if and only if P is O2-doubly continuous. This result demonstrates the equivalence between the O2-convergence being topological and the O2-double continuity of a given poset.

On the other hand, Zhou and Zhao [8] have defined the lim-inf𝓜-convergence in posets to generalize lim-inf-convergence and lim-inf2-convergence [4]. They also found that the lim-inf𝓜-convergence in a poset is topological if and only if the poset is α(𝓜)-continuous when some additional conditions are satisfied (see [8], Theorem 3.1). This result clarified some special conditions of posets under which the lim-inf𝓜-convergence is topological. However, to the best of our knowledge, the equivalent characterization to the lim-inf𝓜-convergence in general posets being topological is still unknown.

One goal of this paper is to propose the notion of 𝓜𝓝-convergence in posets which can unify O-convergence and O2-convergence and search the equivalent characterization to the 𝓜𝓝-convergence being topological. More precisely,

• (G11)

Given a general poset P, we hope to clarify the order-theoretical condition of P which is sufficient and necessary for the 𝓜𝓝-convergence being topological.

• (G12)

Given a poset P satisfying such condition, we hope to provide a topology that can be equipped on P such that the 𝓜𝓝-convergence and the associated topological convergence agree.

Another goal is to look for the equivalent characterization to the lim-inf𝓜-convergence being topological. More precisely,

• (G21)

Given a general poset P, we expect to present a sufficient and necessary condition of P which can precisely serve as an order-theoretical condition for the lim-inf𝓜-convergence being topological.

• (G22)

Given a poset P satisfying such condition, we expect to give a topology on P such that the lim-inf𝓜-convergence and the associated topological convergence are consistent.

To accomplish those goals, motivated by the ideal of introducing the Z-subsets system [9] for defining Z-continuous posets, we propose the notion of 𝓜𝓝-doubly continuous posets and define the 𝓜𝓝-topology on posets in Section 2. Based on the study of the basic properties of the 𝓜𝓝-topology, it is proved that the 𝓜𝓝-convergence in a poset P is topological if and only if P is an 𝓜𝓝-doubly continuous poset if and only if the 𝓜𝓝-convergence and the topological convergence with respect to 𝓜𝓝-topology are consistent. In Section 3, by introducing the notion of α*(𝓜)-continuous posets and presenting the fundamental properties of 𝓜-topology which is induced by the lim-inf𝓜-convergence, we show that the lim-inf𝓜-convergence in a poset P is topological if and only if P is an α*(𝓜)-continuous poset if and only if the lim-inf𝓜-convergence and the topological convergence with respect to 𝓜-topology are consistent.

Some conventional notations will be used in the paper. Given a setX, FX means that F is a finite subset of X. Given a topological space (X, 𝓣) and a net (xi)iI in X, we take $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{T}}{\to }x\end{array}$ to mean the net (xi)iI converges to xP with respect to the topology 𝓣.

Let P be a poset and xP. ↑ x and ↓ x are always used to denote the principal filter {yP : yx} and the principal ideal {zP : zx} of P, respectively. Given a poset P and AP, by writing sup A we mean that the least upper bound of A in P exists and equals to sup AP; dually, by writing inf A we mean that the greatest lower bound of A in P exists and equals to inf AP. And the set A is called an upper set if A = ↑A = {bP; (∃aA) ab}, the lower set is defined dually.

For a poset P, we succinctly denote

• 𝓟(P) = {A : AP}; 𝓟0(P) = 𝓟(P)/{∅};

• 𝓓(P) = {D ∈ 𝓟(P): D is a directed subset of P};

• 𝓕(P) = {F ∈ 𝓟(P): F is a filtered subset of P};

• 𝓛(P) = {L ∈ 𝓟(P): LP}; 𝓛0(P) = 𝓛(P)/{∅};

• 𝓢0(P) = {{x} : xP}.

To make this paper self-contained, we briefly review the following notions:

#### Definition 1.1

([5]). Let P be a poset and x, y, zP. We say y𝓞x if for every net (xi)iI in P which O-converges to xP, xiy holds eventually; dually, we say z𝓞x if for every net (xi)iI in P which O-converges to xP, xiz holds eventually.

#### Definition 1.2

([5]). A poset P is said to be 𝓞-doubly continuous if for every xP, the set {aP : a𝓞x} is directed, the set {bP : b𝓞x} is filtered and sup{aP : a𝓞x} = x = inf{bP : b𝓞x}.

Condition (△). A poset P is said to satisfy Condition(△) if

1. for any x, y, zP, x𝓞yz implies x𝓞z;

2. for any w, s, tP, w𝓞st implies w𝓞t.

#### Definition 1.3

([6]). Let P be a poset and x, y, zP. We say yαx if for every net (xi)iI in P which O2-converges to xP, xiy holds eventually; dually, we say zαx if for every net (xi)iI in P which O2-converges to xP, xiz holds eventually.

#### Definition 1.4

([7]). A poset P is said to be O2-doubly continuous if for every xP,

1. sup{aP : aαx} = x = inf{bP : bαx};

2. for any y, zP with yαx and zαx, there exist A ⊑ {aP : aαx} and B ⊑ {bP : bαx} such that yαc and zαc for each c ∈ ⋂{↑a ∩ ↓b: aA & bB}.

## 2 𝓜𝓝-topology on posets

Based on the introduction of 𝓜𝓝-convergence in posets, the 𝓜𝓝-topology can be defined on posets. In this section, we first define the 𝓜𝓝-double continuity for posets. Then, we show the equivalence between the 𝓜𝓝-convergence being topological and the 𝓜𝓝-double continuity of a given poset.

A PMN-space is a triplet (P, 𝓜,𝓝) which consists of a poset P and two subfamily 𝓜,𝓝 ⊆ 𝓟(P).

All PMN-spaces (P, 𝓜,𝓝) considered in this section are assumed to satisfy the following conditions:

• (C1)

If P has the least element ⊥, then {⊥} ∈ 𝓜;

• (C2)

If P has the greatest element ⊤, then {⊤} ∈ 𝓝;

• (C3)

∅ ∉ 𝓜 and ∅ ∉ 𝓝.

#### Definition 2.1

Let (P, 𝓜,𝓝) be a PMN-space. A net (xi)iI in P is said to 𝓜𝓝-converge to xP if there exist M ∈ 𝓜 and N ∈ 𝓝 satisfying:

• (MN1)

sup M = x = inf N;

• (MN2)

xi ∈ ↑m ∩ ↓n eventually for every mM and every nN.

In this case, we will write $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$.

#### Remark 2.2

Let (P, 𝓜,𝓝) be a PMN-space.

1. If 𝓜 = 𝓓(P) and 𝓝 = 𝓕(P), then a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$P if and only if it O-converges to x. That is to say, O-convergence is a particular case of 𝓜𝓝-convergence.

2. If 𝓜 = 𝓝 = 𝓟0(P), then a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$P if and only if it O2-converges to x. That is to say, O2-convergence is a special case of 𝓜𝓝-convergence.

3. If 𝓜 = 𝓝 = 𝓛0(P), then a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$P if and only if xi = x holds eventually.

4. The 𝓜𝓝-convergent point of a net (xi)iI in P, if exists, is unique.

Indeed, suppose that $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }{x}_{1}\end{array}$ and $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }{x}_{2}.\end{array}$ Then there exist Ak ∈ 𝓜 and Bk ∈ 𝓝 such that sup Ak = xk = inf Bk and akxibk holds eventually for every akAk and bkBk (k = 1, 2). This implies that for any a1A1, a2A2, b1B1 and b2B2, there exists i0I such that a1xi0b2 and a2xi0b1. Thus we have sup A1 = x1 ⩽ inf B2 = x2 and sup A2 = x2 ⩽ inf B1 = x1. Therefore x1 = x2.

5. For any A ∈ 𝓜 and B ∈ 𝓝 with sup A = inf B = xP, we denote $\begin{array}{}{F}_{\left(A,B\right)}^{x}\end{array}$ = {⋂{↑a ∩ ↓b : aA0 & bB0}: A0A & B0B}1. Let $\begin{array}{}{D}_{\left(A,B\right)}^{x}=\left\{\left(d,D\right)\in P×{F}_{\left(A,B\right)}^{x}:d\in D\right\}\end{array}$ and let the preorderon $\begin{array}{}{D}_{\left(A,B\right)}^{x}\end{array}$ be defined by

$(∀(d1,D1),(d2,D2)∈D(A,B)x)(d1,D1)≤(d2,D2)⟺D2⊆D1.$

One can readily check that $\begin{array}{}\left({D}_{\left(A,B\right)}^{x},\le \right)\end{array}$ is directed. Now if we take x(d,D) = d for every $\begin{array}{}\left(d,D\right)\in {D}_{\left(A,B\right)}^{x},\end{array}$, then the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$ because sup A = inf B = x, and ax(d,D)b holds eventually for any aA and bB.

6. Let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\end{array}$ be the net defined in (5) for any A ∈ 𝓜 and B ∈ 𝓝 with sup A = inf B = xP. If $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\end{array}$ converges to pP with respect to some topology 𝓣 on the poset P, then for every open neighborhood Up of p, there exist A0A and B0B such that

$⋂{↑a∩↓b:a∈A0&b∈B0}⊆Up.$

Indeed, suppose that $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\stackrel{\mathcal{T}}{\to }p.\end{array}$ Then for every open neighborhood Up of p, there exists (d0,D0) ∈ $\begin{array}{}{D}_{\left(A,B\right)}^{x}\end{array}$ such that x(d,D) = dUp for all (d, D) ≥ (d0,D0). Since (d, D0) ≥ (d0,D0) for every dD0, x(d,D) = dUp for every dD0. This shows D0Up. So, there exist A0A and B0B such that

$D0=⋂{↑a∩↓b:a∈A0&b∈B0}⊆Up.$

Given a PMN-space (P, 𝓜,𝓝), we can define two new approximate relations $\begin{array}{}{\ll }_{\mathcal{M}}^{\mathcal{N}}\end{array}$ and $\begin{array}{}{⊳}_{\mathcal{M}}^{\mathcal{N}}\end{array}$ on the poset P in the following definition.

#### Definition 2.3

Let (P, 𝓜,𝓝) be a PMN-space and x, y, zP.

1. We define $\begin{array}{}y{\ll }_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ if for any A ∈ 𝓜 and B ∈ 𝓝 with sup A = x = inf B, there exist A0A and B0B such that

$⋂{↑a∩↓b:a∈A0&b∈B0}⊆↑y.$

2. Dually, we define $\begin{array}{}z{⊳}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ if for any M ∈ 𝓜 and N ∈ 𝓝 with sup M = x = inf N, there exist M0M and N0N such that

$⋂{↑m∩↓n:m∈M0&n∈N0}⊆↓z.$

For convenience, given a PMN-space (P, 𝓜,𝓝) and xP, we will briefly denote

• $\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x=\left\{y\in P:y{\ll }_{\mathcal{M}}^{\mathcal{N}}x\right\};\end{array}$

• $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}x=\left\{z\in P:x{\ll }_{\mathcal{M}}^{\mathcal{N}}z\right\};\end{array}$

• $\begin{array}{}{▽}_{\mathcal{M}}^{\mathcal{N}}x=\left\{a\in P:x{⊳}_{\mathcal{M}}^{\mathcal{N}}a\right\};\end{array}$

• $\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x=\left\{b\in P:b{⊳}_{\mathcal{M}}^{\mathcal{N}}x\right\}.\end{array}$

#### Remark 2.4

Let (P, 𝓜,𝓝) be a PMN-space and x, y, zP.

1. If there is no A ∈ 𝓜 such that sup A = x, then $\begin{array}{}p{\ll }_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and $\begin{array}{}p{⊳}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ for all pP; similarly, if there is no B ∈ 𝓝 such that inf B = x, then $\begin{array}{}p{\ll }_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and $\begin{array}{}p{⊳}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ for all pP.

2. By Definition 2.3, one can easily check that if P has the least element ⊥, then $\begin{array}{}\mathrm{\perp }{\ll }_{\mathcal{M}}^{\mathcal{N}}\end{array}$p for every pP, and if P has the greatest element ⊤, then $\begin{array}{}\mathrm{\top }{⊳}_{\mathcal{M}}^{\mathcal{N}}p\end{array}$ for every pP.

3. The implications $\begin{array}{}y{\ll }_{\mathcal{M}}^{\mathcal{N}}x⇒x⩽y\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}z{⊳}_{\mathcal{M}}^{\mathcal{N}}x⇒z⩾x\end{array}$ are not true necessarily. See the following example: letbe the set of all real numbers, in its ordinal order, and 𝓜 = 𝓝 = {{n} : n ∈ ℤ}, whereis the set of all integers. Then, by (1), we have $\begin{array}{}1{\ll }_{\mathcal{M}}^{\mathcal{N}}1/2\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}0{⊳}_{\mathcal{M}}^{\mathcal{N}}1/2.\end{array}$ But 1⧸ ⩽ 1/2 and 0⧸ ⩾ 1/2.

4. Assume that sup A0 = x = inf B0 for some A0 ∈ 𝓜 and B0 ∈ 𝓝. Then it follows from Definition 2.3 that $\begin{array}{}y{\ll }_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ implies yx and $\begin{array}{}z{⊳}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ implies zx. In particular, if 𝓢0(P) ⊆ 𝓜,𝓝, then $\begin{array}{}b{\ll }_{\mathcal{M}}^{\mathcal{N}}a\end{array}$ implies ba and $\begin{array}{}c{⊳}_{\mathcal{M}}^{\mathcal{N}}a\end{array}$ implies ca for any a, b, cP. More particularly, for any p1,p2,p3P, we have $\begin{array}{}{p}_{1}{\ll }_{{\mathcal{S}}_{0}}^{{\mathcal{S}}_{0}}\end{array}$ p2p1p2 and $\begin{array}{}{p}_{3}{⊳}_{{\mathcal{S}}_{0}}^{{\mathcal{S}}_{0}}{p}_{2}\end{array}$p3p2.

#### Proposition 2.5

Let (P, 𝓜,𝓝) be a PMN-space and x, y, zP. Then

1. $\begin{array}{}y{\ll }_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ if and only if for every net (xi)iI that 𝓜𝓝-converges to x, xiy holds eventually.

2. $\begin{array}{}z{⊳}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ if and only if for every net (xi)iI that 𝓜𝓝-converges to x, xiz holds eventually.

#### Proof

(1) Suppose $\begin{array}{}y{\ll }_{\mathcal{M}}^{\mathcal{N}}x\end{array}$. If a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x,\end{array}$ then there exist A ∈ 𝓜 and B ∈ 𝓝 such that sup A = x = inf B, and for any aA and bB, there exists $\begin{array}{}{i}_{a}^{b}\in I\end{array}$ such that axib for all $\begin{array}{}i⩾{i}_{a}^{b}.\end{array}$ According to Definition 2.3 (1), it follows that there exist A0 = {a1,a2, …,an} ⊑ A and B0 = {b1,b2, …,bm} ⊑ B such that x ∈ ⋂{↑ak ∩ ↓bj : 1 ≤ kn & 1 ≤ jm} ⊆ ↑y. Take i0I with that $\begin{array}{}{i}_{0}⩾{i}_{{a}_{k}}^{{b}_{j}}\end{array}$ for every k ∈ {1, 2, …, n} and every j ∈ {1, 2, …, m}. Then xi ∈ ⋂{↑ak ∩ ↓bj : 1 ≤ kn & 1 ≤ jm} ⊆ ↑y for all ii0. This means xiy holds eventually.

Conversely, suppose that for every net (xi)iI that 𝓜𝓝-converges to x, xiy holds eventually. For every A ∈ 𝓜 and B ∈ 𝓝 with sup A = x = inf B, consider the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\end{array}$ defined in Remark 2.2 (5). By Remark 2.2 (5), the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\stackrel{\mathcal{M}\mathcal{N}}{\to }x.\end{array}$ So, there exists (d0,D0) ∈ $\begin{array}{}{D}_{\left(A,B\right)}^{x}\end{array}$ such that x(d,D) = dy for all (d, D) ≥ (d0,D0). Since (d, D0) ≥ (d0,D0) for all dD0, x(d,D0) = dy for all dD0. Thus, we have D0 ⊆ ↑y. It follows from the definition of $\begin{array}{}{D}_{\left(A,B\right)}^{x}\end{array}$ that there exist A0A and B0B such that D0 = ⋂{↑a ∩ ↓b : aA0 & bB0} ⊆ ↑y. This shows $\begin{array}{}y{\ll }_{\mathcal{M}}^{\mathcal{N}}x.\end{array}$

The proof of (2) can be processed similarly. □

#### Remark 2.6

Let (P, 𝓜,𝓝) be a PMN-space.

1. If 𝓜 = 𝓓(P) and 𝓝 = 𝓕(P), then $\begin{array}{}{\ll }_{\mathcal{D}}^{\mathcal{F}}={\ll }_{\mathcal{O}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{⊳}_{\mathcal{D}}^{\mathcal{F}}={⊳}_{\mathcal{O}}.\end{array}$

2. If 𝓜 = 𝓝 = 𝓟0(P), then $\begin{array}{}{\ll }_{{\mathcal{P}}_{0}}^{{\mathcal{P}}_{0}}={\ll }_{\alpha }\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and{⊳}_{{\mathcal{P}}_{0}}^{{\mathcal{P}}_{0}}={⊳}_{\alpha }.\end{array}$

Given a PMN-space (P, 𝓜,𝓝), depending on the approximate relations $\begin{array}{}{\ll }_{\mathcal{M}}^{\mathcal{N}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{⊳}_{\mathcal{M}}^{\mathcal{N}}\end{array}$ on P. we can define the 𝓜𝓝-double continuity for the poset P.

#### Definition 2.7

Let (P, 𝓜,𝓝) be a PMN-space. The poset P is called an 𝓜𝓝-doubly continuous poset if for every xP, there exist Mx ∈ 𝓜 and Nx ∈ 𝓝 such that

• (A1)

Mx$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x,{N}_{x}\subseteq {△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and sup Mx = x = inf Nx.

• (A2)

For any $\begin{array}{}y\in {▾}_{\mathcal{M}}^{\mathcal{N}}x\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}z\in {△}_{\mathcal{M}}^{\mathcal{N}}x,\end{array}$ ⋂{↑m ∩ ↓n : mM0 & nN0} ⊆ $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\end{array}$ for some M0Mx and N0Nx.

By Remark 2.4 (4) and Definition 2.7, we have the following basic property about 𝓜𝓝-doubly continuous posets:

#### Proposition 2.8

Let (P, 𝓜,𝓝) be a PMN-space and x, y, zP. If the poset P is an 𝓜𝓝-doubly continuous poset, then $\begin{array}{}y{\ll }_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ implies yx and $\begin{array}{}z{⊳}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ implies zx.

#### Example 2.9

Let (P, 𝓜,𝓝) be a PMN-space.

1. If 𝓜 = 𝓝 = 𝓢0(P), then by Remark 2.4 (4), we have $\begin{array}{}{\ll }_{{\mathcal{S}}_{0}}^{{\mathcal{S}}_{0}}=⩽\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{⊳}_{{\mathcal{S}}_{0}}^{{\mathcal{S}}_{0}}=⩾.\end{array}$ By Definition 2.7, one can easily check that P is an 𝓢0𝓢0-doubly continuous poset.

2. If 𝓜 = 𝓝 = 𝓛0(P), then by Definition 2.3, we have $\begin{array}{}{\ll }_{{\mathcal{L}}_{0}}^{{\mathcal{L}}_{0}}=⩽\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{⊳}_{{\mathcal{L}}_{0}}^{{\mathcal{L}}_{0}}=⩾.\end{array}$ It can be easily checked from Definition 2.7 that P is an 𝓛0𝓛0-doubly continuous poset.

3. Let 𝓜 = 𝓓(P) and 𝓝 = 𝓕(P). Then it is easy to check that if P is an 𝓞-doubly continuous poset which satisfies Condition (), then it is a 𝓓𝓕-doubly continuous poset. Particularly, finite posets, chains and anti-chains, completely distributive lattices are all 𝓓𝓕-doubly continuous posets.

4. Let 𝓜 = 𝓝 = 𝓟0(P). Then the poset P is 𝓟0𝓟0-double continuous if and only if it is O2-double continuous. Thus, chains and finite posets are all 𝓟0𝓟0-doubly continuous posets.

Next, we are going to consider the 𝓜𝓝-topology on posets, which is induced by the 𝓜𝓝-convergence.

#### Definition 2.10

Given a PMN-space (P, 𝓜,𝓝), a subset U of P is called an 𝓜𝓝-open set if for every net (xi)iI with that $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$U, xiU holds eventually.

Clearly, the family $\begin{array}{}{\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right)\end{array}$ consisting of all 𝓜𝓝-open subsets of P forms a topology on P. And this topology is called the 𝓜𝓝-topology.

#### Theorem 2.11

Let (P, 𝓜,𝓝) be a PMN-space. Then a subset U of P is an 𝓜𝓝-open set if and only if for every M ∈ 𝓜 and N ∈ 𝓝 with sup M = x = inf NU, we have

$⋂{↑m∩↓n:m∈M0&n∈N0}⊆U$

for some M0M and N0N.

#### Proof

Suppose that U is an 𝓜𝓝-open subset of P. For every M ∈ 𝓜 and N ∈ 𝓝 with sup M = x = inf NU, let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(M,N\right)}^{x}}\end{array}$ be the net defined in Remark 2.2 (5). Then the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(M,N\right)}^{x}}\stackrel{\mathcal{M}\mathcal{N}}{\to }x.\end{array}$ By the definition of 𝓜𝓝-open set, the exists (d0,D0) ∈ $\begin{array}{}{D}_{\left(M,N\right)}^{x}\end{array}$ such that x(d,D) = dU for all (d, D) ≥ (d0,D0). Since (d, D0) ≥ (d0,D0) for all dD0, x(d,D0) = dU for every dD0, and thus D0U. It follows from the definition of the directed set $\begin{array}{}{D}_{\left(M,N\right)}^{x}\end{array}$ that D0 = ⋂{↑m ∩ ↓n : mM0 & nN0} ⊆ U for some M0M and some N0N.

Conversely, assume that U is a subset of P with the property that for any M ∈ 𝓜 and N ∈ 𝓝 with sup M = x = inf NU, there exist M0 = {m1,m2, …,mk} ⊑ M and N0 = {n1,n2, …,nl} ⊑ N such that ⋂{↑mh ∩ ↓nj : 1 ≤ hk & 1 ≤ jl} ⊆ U. Let (xi)iI be a net that 𝓜𝓝-converges to xU. Then there exist M ∈ 𝓜 and N ∈ 𝓝 such that sup M = x = inf NU, and for every mM and nN, mxin holds eventually. This means that for every mhM0 and njN0, there exists ih,jI such that mhxinj for all iih,j. Take i0I such that i0ih,j for all h ∈ {1, 2, …, k} and j ∈ {1, 2, …, l}. Then xi ∈ ⋂{↑mh ∩ ↓nj : 1 ≤ hk & 1 ≤ jl} ⊆ U for all ii0. Therefore, U is an 𝓜𝓝-open subset of P. □

#### Proposition 2.12

Let (P, 𝓜,𝓝) be a PMN-space in which P is an 𝓜𝓝-doubly continuous poset, and y, zP. Then $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\in {\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right).\end{array}$

#### Proof

Suppose that M ∈ 𝓜 and N ∈ 𝓝 with sup M = inf N = x$\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z.\end{array}$ Since P is an 𝓜𝓝-doubly continuous poset, there exist Mx ∈ 𝓜 and Nx ∈ 𝓝 satisfying condition (A1) and (A2) in Definition 2.7. This means that there exist M0Mx$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and N0Nx$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ such that ⋂{↑m0 ∩ ↓n0:m0M0 & n0N0} ⊆ $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z.\end{array}$ As M0Mx$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and N0Nx$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$, by Definition 2.3, there exist Mm0M and Nn0N such that ⋂{↑m ∩ ↓n : mMm0 & nNn0 } ⊆ ↑m0 ∩ ↓n0 for every m0M0 and n0N0. Take MF = ⋃{Mm0:m0M0} and NF = ⋃{Nn0:n0N0}. Then it is easy to check that MFM, NFN and

$x∈⋂{↑a∩↓b:a∈MF&b∈NF}⊆⋂{↑m0∩↓n0:m0∈M0&n0∈N0}⊆▴MNy∩▽MNz.$

So, it follows from Theorem 2.11 that $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\in {\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right).\end{array}$ □

#### Lemma 2.13

Let (P, 𝓜,𝓝) be a PMN-space in which P is an 𝓜𝓝-doubly continuous poset. Then a net

$(xi)i∈I→MNx∈P⟺(xi)i∈I→OMN(P)x.$

#### Proof

From the definition of $\begin{array}{}{\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right),\end{array}$ it is easy to see that a net

$(xi)i∈I→MNx∈P⟹(xi)i∈I→OMN(P)x.$

To prove the Lemma, it suffices to show that a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{{\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right)}{\to }x\end{array}$P implies $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$. Suppose a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{{\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right)}{\to }x\end{array}$. Since P is an 𝓜𝓝-doubly continuous poset, there exist Mx ∈ 𝓜 and Nx ∈ 𝓝 such that Mx$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$, Nx$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and sup Mx = x = inf Nx. By Proposition 2.12, $\begin{array}{}x\in {▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\in {\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right)\end{array}$ for every yMx$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and every zNx$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$, and hence $\begin{array}{}{x}_{i}\in {▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\end{array}$ holds eventually for every yMx$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and every zNx$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$. It follows from Proposition 2.8 that yxiz holds eventually for every yMx and zNx. Thus $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$ □

#### Lemma 2.14

Let (P, 𝓜,𝓝) be a PMN-space. If the 𝓜𝓝-convergence in P is topological, then P is 𝓜𝓝-doubly continuous.

#### Proof

Suppose that the 𝓜𝓝-convergence in P is topological. Then there exists a topology 𝓣 on P such that for every xP, a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$ if and only if $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{T}}{\to }x\end{array}$. Define Ix = {(p, U) ∈ P × 𝓝(x) : pU}, where 𝓝(x) denotes the set of all open neighbourhoods of x in the topological space (P, 𝓣), i.e., 𝓝(x) = {U ∈ 𝓣 : xU}. Define the preorder ≼ on Ix as follows:

$(∀(p1,U1),(p2,U2)∈Ix)(p1,U1)≼(p2,U2)⟺U2⊆U1.$

Now one can easily see that Ix is directed. Let x(p,U) = p for every (p, U) ∈ Ix. Then it is straightforward to check that the net $\begin{array}{}\left({x}_{\left(p,U\right)}{\right)}_{\left(p,U\right)\in {I}_{x}}\stackrel{\mathcal{T}}{\to }x,\end{array}$ and thus $\begin{array}{}\left({x}_{\left(p,U\right)}{\right)}_{\left(p,U\right)\in {I}_{x}}\stackrel{\mathcal{M}\mathcal{N}}{\to }x.\end{array}$ By Definition 2.1, there exist Mx ∈ 𝓜 and Nx ∈ 𝓝 such that sup Mx = x = inf Nx, and for every mMx and nNx, there exists $\begin{array}{}\left({p}_{m}^{n},{U}_{m}^{n}\right)\end{array}$Ix such that x(p,U) = p ∈ ↑m ∩ ↓n for all $\begin{array}{}\left(p,U\right)\succcurlyeq \left({p}_{m}^{n},{U}_{m}^{n}\right).\end{array}$ Since $\begin{array}{}\left(p,{U}_{m}^{n}\right)\succcurlyeq \left({p}_{m}^{n},{U}_{m}^{n}\right)\end{array}$ for every $\begin{array}{}p\in {U}_{m}^{n},\end{array}$ $\begin{array}{}{x}_{\left(p,{U}_{m}^{n}\right)}=p\end{array}$ ∈ ↑m ∩ ↓n for every $\begin{array}{}p\in {U}_{m}^{n}.\end{array}$ This shows

$(∀m∈Mx,n∈Nx)(∃Umn∈N(x))x∈Umn⊆↑m∩↓n.$(*)

For any A ∈ 𝓜 and B ∈ 𝓝 with sup A = x = inf B, let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\end{array}$ be the net defined as in Remark 2.2 (5). Then $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\stackrel{\mathcal{M}\mathcal{N}}{\to }x,\end{array}$ and hence $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left(A,B\right)}^{x}}\stackrel{\mathcal{T}}{\to }x.\end{array}$ This implies, by Remark 2.2 (6), that there exist A0A and B0B satisfying

$x∈⋂{↑a∩↓b:a∈A0&b∈B0}⊆Umn⊆↑m∩↓n.$

Therefore, m$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and n$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$, and hence $\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and Nx$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$.

Let y$\begin{array}{}{▾}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$ and z$\begin{array}{}{△}_{\mathcal{M}}^{\mathcal{N}}x\end{array}$. Since sup Mx = x = inf Nx, by Definition 2.3, ⋂{↑m ∩ ↓n : mM1 & nN1} ⊆ ↑y ∩ ↓z for some M1Mx and N1Nx. This concludes by Condition (⋆) and the finiteness of sets M1 and N1 that $\begin{array}{}\bigcap \left\{{U}_{m}^{n}:m\in {M}_{1}\phantom{\rule{thinmathspace}{0ex}}\mathrm{&}\phantom{\rule{thinmathspace}{0ex}}n\in {N}_{1}\right\}\end{array}$ ∈ 𝓝(x) and

$x∈⋂{Umn:m∈M1&n∈N1}⊆⋂{↑m∩↓n:m∈M1&n∈N1}⊆↑y∩↓z.$

Considering the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left({M}_{x},{N}_{x}\right)}^{x}}\end{array}$ defined in Remark 2.2 (5), we have $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left({M}_{x},{N}_{x}\right)}^{x}}\stackrel{\mathcal{M}\mathcal{N}}{\to }x,\end{array}$ and hence $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left({M}_{x},{N}_{x}\right)}^{x}}\stackrel{\mathcal{T}}{\to }x.\end{array}$ So, by Remark 2.2 (6), there exist M2Mx and N2Nx such that

$x∈⋂{↑m∩↓n:m∈M2&n∈N2}⊆⋂{Umn:m∈M1&n∈N1}⊆↑y∩↓z.$

Finally, we show ⋂{↑m ∩ ↓n : mM2 & nN2} ⊆ $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\end{array}$. Let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left({M}^{\prime },{N}^{\prime }\right)}^{{x}^{\prime }}}\end{array}$ be the net defined in 2.2 (5) for any M′ ∈ 𝓜 and N′ ∈ 𝓝 with sup M′ = inf N′ = x′ ∈ ⋂{↑m ∩ ↓n : mM2 & nN2}. Then $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left({M}^{\prime },{N}^{\prime }\right)}^{{x}^{\prime }}}\stackrel{\mathcal{M}\mathcal{N}}{\to }{x}^{\prime },\end{array}$ and thus $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{\left({M}^{\prime },{N}^{\prime }\right)}^{{x}^{\prime }}}\stackrel{\mathcal{T}}{\to }{x}^{\prime }.\end{array}$ This implies by Remark 2.2 (6) that there exist $\begin{array}{}{M}_{0}^{\prime }⊑{M}^{\prime }\end{array}$ and $\begin{array}{}{N}_{0}^{\prime }⊑{N}^{\prime }\end{array}$ satisfying

$x′∈⋂{↑m′∩↓n′:m∈M0′&n∈N0′}⊆⋂{Umn:m∈M1&n∈N1}⊆↑y∩↓z.$

Hence, we have x′ ∈ $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\end{array}$ by Definition 2.3. This shows ⋂{↑m ∩ ↓n : mM2 & nN2} ⊆ $\begin{array}{}{▴}_{\mathcal{M}}^{\mathcal{N}}y\cap {▽}_{\mathcal{M}}^{\mathcal{N}}z\end{array}$. Therefore, it follows from Definition 2.7 that P is 𝓜𝓝-doubly continuous. □

Combining Lemma 2.13 and Lemma 2.14, we obtain the following theorem.

#### Theorem 2.15

Let (P, 𝓜,𝓝) be a PMN-space. Then the following statements are equivalent:

1. P is an 𝓜𝓝-doubly continuous poset.

2. For any net (xi)iI in P, $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}\mathcal{N}}{\to }x\end{array}$ if and only if $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{{\mathcal{O}}_{\mathcal{M}}^{\mathcal{N}}\left(P\right)}{\to }x\end{array}$.

3. The 𝓜𝓝-convergence in P is topological.

#### Proof

(1) ⇒ (2): By Lemma 2.13.

(2) ⇒ (3): It is clear.

(3) ⇒ (1): By Lemma 2.14. □

## 3 𝓜-topology induced by lim-inf𝓜-convergence

In this section, the notion of lim-inf𝓜-convergence is reviewed and the 𝓜-topology on posets is defined. By exploring the fundamental properties of the 𝓜-topology, those posets under which the lim-inf𝓜-convergence is topological are precisely characterized.

By saying a PM-space, we mean a pair (P, 𝓜) that contains a poset P and a subfamily 𝓜 of 𝓟(P).

#### Definition 3.1

([8]). Let (P, 𝓜) be a PM-space. A net (xi)iI in P is said to lim-inf𝓜-converge to xP if there exists M ∈ 𝓜 such that

• (M1)

x ⩽ sup M;

• (M2)

for every mM, xim holds eventually.

In this case, we write $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }x\end{array}$.

It is worth noting that both lim-inf-convergence and lim-inf2-convergence [4] in posets are particular cases of lim-inf𝓜-convergence.

#### Remark 3.2

Let (P, 𝓜) be a PM-space and x, yP.

1. Suppose that a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }x\end{array}$ and yx. $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }y\end{array}$ by Definition 3.1. This concludes that the set of all lim-inf𝓜-convergent points of the net (xi)iI in P is a lower subset of P. Thus, the lim-inf𝓜-convergent points of the net (xi)iI need not be unique.

2. If P has the least elementand ∅ ∈ 𝓜, then we have $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }\mathrm{\perp }\end{array}$ for every net (xi)iI in P.

3. For every M ∈ 𝓜 with sup Mx, we denote $\begin{array}{}{F}_{M}^{x}\end{array}$ = {⋂{↑m : mM0} : M0M}2. Let $\begin{array}{}{D}_{M}^{x}\end{array}$ = {(d, D) ∈ P × $\begin{array}{}{F}_{M}^{x}\end{array}$ : dD} be in the preorderdefined by

$(∀(d1,D1),(d2,D2)∈DMx)(d1,D1)≦(d2,D2)⟺D2⊆D1.$

It is easy to see that the set $\begin{array}{}{D}_{M}^{x}\end{array}$ is directed. Take x(d,D) = d for every (d, D) ∈ $\begin{array}{}{D}_{M}^{x}\end{array}$. Then, by Definition 3.1, one can straightforwardly check that the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\stackrel{\mathcal{M}}{\to }\end{array}$ for every ax.

4. If the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\end{array}$ defined in (3) converges to pP with respect to some topology 𝓣 on P, then for every open neighbourhood Up of p, there exists M0M such that ⋂{↑m : mM0} ⊆ Up.

#### Definition 3.3

([8]). Let (P, 𝓜) be a PM-space.

1. For x, yP, define yα(𝓜)x if for every net (xi)iI that lim-inf𝓜-converges to x, xiy holds eventually.

2. The poset P is said to be α(𝓜)-continuous if {xP : xα(𝓜)a} ∈ 𝓜 and a = sup{xP : xα(𝓜)a} holds for every aP.

Given a PM-space (P, 𝓜), the approximate relation ≪α(𝓜) on the poset P can be equivalently characterized in the following proposition.

#### Proposition 3.4

Let (P, 𝓜) be a PM-space and x, yP. Then yα(𝓜)x if and only if for every M ∈ 𝓜 with sup Mx, there exists M0M such that

$⋂{↑m:m∈M0}⊆↑y.$

#### Proof

Suppose yα(𝓜)x. Let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\end{array}$ be the net defined in Remark 3.2 (3) for every M ∈ 𝓜 with sup M = px. Then the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\stackrel{\mathcal{M}}{\to }x.\end{array}$ By Definition 3.3 (1), there exists (d0,D0) ∈ $\begin{array}{}{D}_{M}^{x}\end{array}$ such that x(d,D) = dy for all (d, D) ≦ (d0,D0). Since (d, D0) ≦ (d0,D0) for every dD0, x(d,D0) = dy for every dD0. So D0 ⊆ ↑y. This shows that there exists M0M such that D0 = ⋂{↑m : mM0} ⊆ ↑y.

Conversely, suppose that for every M ∈ 𝓜 with sup Mx, there exists M0M such that ⋂{↑m : mM0} ⊆ ↑y. Let (xi)iI be a net that lim-inf𝓜-converges to x. Then, by Definition 3.1, there exists M ∈ 𝓜 such that sup M = px, and for every mM, there exists imI such that xim for all iim. Take i0I with that i0im for every mM0M, we have that xi ∈ ⋂{↑m : mM0} ⊆ ↑y for all ii0. This shows that xiy holds eventually. Thus, by Definition 3.3 (1), we have yα(𝓜)x. □

#### Remark 3.5

Let (P, 𝓜) be a PM-space and x, yP.

1. If there is no M ∈ 𝓜 such that sup Mx, then pα(𝓜)x for every pP. And, if the poset P has the least element ⊥, then ⊥≪α(𝓜)p for every pP.

2. The implication yα(𝓜)xyx may not be true. For example, let P = {0,1, 2, …} be in the discrete orderdefined by

$(∀i,j∈P)i⩽j⟺i=j.$

And let 𝓜 = {{2}}. Then, it is easy to see from Remark 3.5 (1) that 0≪α(𝓜)1 and 0⧸ ⩽ 1.

3. Assume the PM-space (P, 𝓜) has the property that for every pP, there exists Mp ∈ 𝓜 such that sup Mp = p. Then, by Proposition 3.4, we have

$(∀q,r∈P)q≪α(M)r⟹q⩽r.$

For more interpretations of the approximate relation ≪α(𝓜) on posets, the readers can refer to Example 3.2 and Remark 3.3 in [8].

For simplicity, given a PM-space (P, 𝓜) and xP, we will denote

• 𝓜x = {yP : yα(𝓜)x};

• 𝓜x = {zP : xα(𝓜)z}.

Based on the approximate relation ≪α(𝓜) on posets, the α*(𝓜)-continuity can be defined for posets in the following:

#### Definition 3.6

Let (P, 𝓜) be a PM-space. The poset P is called an α*(𝓜)-continuous poset if for every xP, there exists Mx ∈ 𝓜 such that

• (O1)

sup Mx = x and Mx ⊆ ▾𝓜x. And,

• (O1)

for every y ∈ ▾𝓜x, there exists FMx such that ⋂{↑f : fF} ⊆ ▴𝓜y.

Noticing Remark 3.5 (3), we have the following proposition about α*(𝓜)-continuous posets.

#### Proposition 3.7

Let (P, 𝓜) be a PM-space in which the poset P is α*(𝓜)-continuous. Then

$(∀x,y∈P)y≪α(M)x⟹y⩽x.$

The following examples of α*(𝓜)-continuous posets can be formally checked by Definition 3.6.

#### Example 3.8

Let (P, 𝓜) be a PM-space.

1. If P is a finite poset, then P is an α*(𝓜)-continuous poset if and only if for every xP, there exists Mx ∈ 𝓜 such that sup Mx = x.

2. Let 𝓜 = 𝓛(P). Then P is an α*(𝓛)-continuous poset. This means that every poset is α*(𝓛)-continuous.

3. Let 𝓜 = 𝓓(P). Then we have ≪ = ≪α(𝓓) (see Example 3.2 (1) in [8]). The poset P is a continuous poset if and only if it is an α*(𝓓)-continuous poset. In particular, finite posets, chains, anti-chains and completely distributive lattices are all α*(𝓓)-continuous.

4. Let 𝓜 = 𝓟(P). If P is a finite poset (resp. chain, anti-chain), then P is an α*(𝓟)-continuous poset.

#### Proposition 3.9

Let (P, 𝓜) be a PM-space. If P is an α(𝓜)-continuous poset, and {yP : (∃ zP) yα(𝓜)z≪α(𝓜)a} ∈ 𝓜 for every aP, then P is an α*(𝓜)-continuous poset.

#### Proof

Suppose that P is an α(𝓜)-continuous poset, and {yP : (∃ zP) yα(𝓜)z≪α(𝓜)a} ∈ 𝓜 for every aP. Take Ma = ▾𝓜a. Then it is easy to see that sup Ma = a and Ma ⊆ ▾𝓜a. By Remark 3.3 (4) in [8], we have sup{yP : (∃ zP) yα(𝓜)z≪α(𝓜)a} = a. This implies, by Proposition 3.4 and Remark 3.5 (2), that for every y ∈ ▾𝓜a, there exist {y1,y2, …,yn}, {z1,z2, …,zn} ⊑ Ma = ▾𝓜a such that

$⋂{↑zi:i∈{1,2,…,n}}⊆⋂{↑yi:i∈{1,2,…,n}}⊆↑y,$

and yiα(𝓜)ziα(𝓜)a for every i ∈ {1, 2, …, n}. Next, we show ⋂{↑zi : i ∈ {1, 2, …, n}} ⊆ ▴𝓜y. For every M ∈ 𝓜 with sup Mb ∈ ⋂{↑zi : i ∈ {1, 2, …, n}}, by Proposition 3.4, there exists MiM such that ⋂{↑m′:m′ ∈ Mi} ⊆ ↑yi for every i ∈ {1, 2, …, n}. Take M0 = ⋃{Mi : i ∈ {1, 2, …, n}}. Then M0M and

$⋂{↑m:m∈M0}⊆⋂{↑yi:i∈{1,2,…,n}}⊆↑y.$

This shows yα(𝓜)b for every b ∈ ⋂{↑zi : i ∈ {1, 2, …, n}}. Hence, ⋂{↑zi : i ∈ {1, 2, …, n}} ⊆ ▴𝓜y. Thus P is an α*(𝓜)-continuous poset. □

The fact that an α*(𝓜)-continuous poset P in a PM-space (P, 𝓜) may not be α(𝓜)-continuous can be demonstrated in the following example.

#### Example 3.10

Let (P, 𝓜) be the PM-space in which the poset P = ℝ is the set of all real number with its usual orderand 𝓜 = 𝓢0(ℝ). Then we haveα(𝓢0) = ⩽ by Proposition 3.4. It is easy to check, by Definition 3.6, thatis an α*(𝓢0)-continuous poset. Butis not an α(𝓢0)-continuous poset because𝓢0x = ↓x⧸ ∈ 𝓢0(P) for every x ∈ ℝ.

We turn to consider the topology induced by the lim-inf𝓜-convergence in posets.

#### Definition 3.11

Let (P, 𝓜) be a PM-space. A subset V of P is said to be 𝓜-open if for every net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }x\in V,\end{array}$ xiV holds eventually.

Given a PM-space (P, 𝓜), one can formally verify that the set of all 𝓜-open subsets of P forms a topology on P. This topology is called the 𝓜-topology, and denoted by 𝓞𝓜(P).

The following Theorem is an order-theoretical characterization of 𝓜-open sets.

#### Theorem 3.12

Let (P, 𝓜) be a PM-space. Then a subset V of P is 𝓜-open if and only if it satisfies the following two conditions:

• (V1)

V = V, i.e., V is an upper set.

• (V2)

For every M ∈ 𝓜 with sup MV, there exists M0M such that ⋂{↑m : mM0} ⊆ V.

#### Proof

Suppose that V is an 𝓜-open subset of P. By Remark 3.2 (1), it is easy to see that V is an upper set. Let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\end{array}$ be the net defined in Remark 3.2 (3) for every M ∈ 𝓜 with sup M = xV. Then $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\stackrel{\mathcal{M}}{\to }\phantom{\rule{thinmathspace}{0ex}}x\phantom{\rule{thinmathspace}{0ex}}\in \phantom{\rule{thinmathspace}{0ex}}V.\end{array}$ This implies, by Definition 3.11, that there exists (d0,D0) ∈ $\begin{array}{}{D}_{M}^{x}\end{array}$ such that x(d,D) = dV for all (d, D) ≥ q (d0,D0). Since (d, D0) ≦ (d0,D0) for all dD0, x(d,D0) = dV for all dD0. This shows D0V. Thus there exists M0M such that D0 = ⋂{↑m : mM0} ⊆ V.

Conversely, suppose V is a subset of P which satisfies Condition (V1) and (V2). Let (xi)iI be a net that lim-inf𝓜-converges to xV. Then there exists M ∈ 𝓜 such that sup M = yxV = ↑V (hence, yV), and for every mM, there exists imI such that xim for all iim. By Condition (V2), we have that ⋂{↑m : mM0} ⊆ V for some M0M. Take i0I with that i0im for all mM0. Then xi ∈ ⋂{↑m : mM0} ⊆ V for all ii0. This shows that V is an 𝓜-open set. □

Recall that given a topological space (X, 𝓣) and a point xP, a family 𝓑(x) of open neighbourhoods of x is called a base for the topological space (X, 𝓣) at the point x if for every neighbourhood V of x there exists an U ∈ 𝓑(x) such that xUV.

If the poset P in a PM-space (P, 𝓜) is an α*(𝓜)-continuous poset, we provide a base for the topological space (P, 𝓞𝓜(P)) at a point xP.

#### Proposition 3.13

Let (P, 𝓜) be a PM-space in which the poset P is α*(𝓜)-continuous. Then𝓜x ∈ 𝓞𝓜(P) for every xP.

#### Proof

One can readily see, by Proposition 3.4, that {▴𝓜}x is an upper subset of P for every xP. For every M ∈ 𝓜 with sup M = y ∈ {▴𝓜}x, by Definition 3.6 (O1) there exists My ∈ 𝓜 such that My ⊆ {▾𝓜}y and sup My = y. Since xα(𝓜)y, by Definition 3.6 (O2), we have ⋂{↑mi : i ∈ {1, 2, …, n}} ⊆ {▴𝓜}x for some {m1,m2, …,mn} ⊑ My. Observing {m1,m2, …,mn} ⊑ My ⊆ {▾𝓜}y, we can conclude that there exists MiM such that ⋂{↑a : aMi} ⊆ ↑mi for every i ∈ {1, 2, …, n}. Let M0 = ⋃{Mi : i ∈ {1, 2, …, n}}. Then M0M and

$⋂{↑m:m∈M0}⊆⋂{↑mi:i∈{1,2,…,n}}⊆▴Mx.$

This shows, by Theorem 3.12, that ▴𝓜x ∈ 𝓞𝓜(P) for every xP. □

#### Proposition 3.14

Let (P, 𝓜) be a PM-space in which the poset P is α*(𝓜)-continuous and xP. Then {⋂{▴𝓜a : aA} : A ⊑ ▾𝓜x} is a base for the topological space (P, 𝓞𝓜(P)) at the point x.

#### Proof

Clearly, by Proposition 3.13, we have ⋂{▴𝓜a : aA} ∈ 𝓞𝓜(P) for every A ⊑ ▾𝓜x. Let U ∈ 𝓞𝓜(P) and xU. Since P is an α*(𝓜)-continuous poset, there exists Mx ∈ 𝓜 such that Mx ⊆ ▾𝓜x and sup Mx = xU. By Theorem 3.12, it follows that ⋂{↑m : mM0} ⊆ U for some M0Mx ⊆ ▾𝓜x. So, from Proposition 3.7, we have

$x∈⋂{▴Mm:m∈M0}⊆⋂{↑m:m∈M0}⊆U.$

Thus, {⋂{▴𝓜a : aA} : A ⊑ ▾𝓜x} is a base for the topological space (P, 𝓞𝓜(P)) at the point x. □

In the rest, we are going to establish a characterization theorem which demonstrates the equivalence between the lim-inf𝓜-convergence being topological and the α*(𝓜)-continuity of the poset in a given PM-space.

#### Lemma 3.15

Let (P, 𝓜) be a PM-space. If P is an α*(𝓜)-continuous poset, then a net

$(xi)i∈I→Mx∈P⟺(xi)i∈I→OM(P)x.$

#### Proof

By the definition of 𝓞𝓜(P), it is easy to see that a net

$(xi)i∈I→Mx∈P⟹(xi)i∈I→OM(P)x.$

To prove the Lemma, we only need to show that a net $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{{\mathcal{O}}_{\mathcal{M}}\left(P\right)}{\to }x\in P\end{array}$ implies $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }x.\end{array}$ Suppose $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{{\mathcal{O}}_{\mathcal{M}}\left(P\right)}{\to }x.\end{array}$ As P is an α*(𝓜)-continuous poset, there exists Mx ∈ 𝓜 such that Mx ⊆ ▾𝓜x and sup Mx = x. By Proposition 3.13, we have x ∈ ▴𝓜y ∈ {𝓞𝓜(P)} for every yMx ⊆ ▾𝓜x. Hence, xi ∈ ▴𝓜y holds eventually. This implies, by Proposition 3.7, that xi ∈ ▴𝓜y ⊆ ↑y holds eventually. By the definition of lim-inf𝓜-convergence, we have $\begin{array}{}\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }x.\end{array}$ □

In the converse direction, we have the following Lemma.

#### Lemma 3.16

Let (P, 𝓜) be a PM-space. If the lim-inf𝓜-convergence in P is topological, then P is an α*(𝓜)-continuous poset.

#### Proof

Suppose that the lim-inf𝓜-convergence in P is topological. Then there exists a topology 𝓣 such that for every xP, a net

$(xi)i∈I→Mx⟺(xi)i∈I→Tx.$

Define Ix = {(p, V) ∈ P × 𝓝(x) : pV}, where 𝓝(x) is the set of all open neighbourhoods of x, namely, 𝓝(x) = {V ∈ 𝓣 : xV}. Define also the preorder ⪯ on Ix as follows:

$(∀(p1,V1),(p2,V2)∈Ix)(p1,V1)⪯(p2,V2)⟺V2⊆V1.$

It is easy to see that Ix is directed. Now, let x(p,V) = p for every (p, V) ∈ Ix. Then one can readily check that the net $\begin{array}{}\left({x}_{\left(p,V\right)}{\right)}_{\left(p,V\right)\in {I}_{x}}\stackrel{\mathcal{T}}{\to }x,\end{array}$ and hence $\begin{array}{}\left({x}_{\left(p,V\right)}{\right)}_{\left(p,V\right)\in {I}_{x}}\stackrel{\mathcal{M}}{\to }x.\end{array}$ This means that there exists Mx ∈ 𝓜 such that sup Mxx, and for every mMx, there exists (pm,Vm) ∈ Ix with that x(p,V) = pm for all (p, V) ⪰ (pm,Vm). Since (p, Vm) ⪰ (pm,Vm) for all pVm, we have x(p,Vm) = pm for all pVm. This shows

$(∀m∈Mx)(∃Vm∈N(x))x∈Vm⊆↑m.$(⋆⋆)

Next we prove Mx ⊆ ▾𝓜x. For every mMx and every MM with sup Mx, let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\end{array}$ be the net defined in Remark 3.2 (3). Then the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\stackrel{\mathcal{M}}{\to }x,\end{array}$ and thus $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{M}^{x}}\stackrel{\mathcal{T}}{\to }x.\end{array}$ It follows from Remark 3.2 (4) that there exists M0M such that x ∈ ⋂{↑a : aM0} ⊆ Vm. By Condition (⋆⋆), we have x ∈ ⋂{↑a : aM0} ⊆ Vm ⊆ ↑m. So, mα(𝓜)x. This shows Mx ⊆ ▾𝓜x.

Let y ∈ ▾𝓜x. Then there exists {m1,m2, …,mn} ⊑ Mx such that ⋂{↑mi : i ∈ {1, 2, …, n}} ⊆ ↑y as Mx ∈ 𝓜 and sup Mxx. By Condition (⋆⋆), it follows that ⋂Vmi : i ∈ {1, 2, …, n}} ⊆ ⋂{↑mi : i ∈ {1, 2, …, n}} ⊆ ↑y. Considering the net $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{{M}_{x}}^{x}}\end{array}$ defined in Remark 3.2 (3), we have $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{{M}_{x}}^{x}}\stackrel{\mathcal{M}}{\to }\end{array}$ x, and hence $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{{M}_{x}}^{x}}\stackrel{\mathcal{T}}{\to }x.\end{array}$ This implies, by Remark 3.2 (4), that

$⋂{↑b:b∈M00}⊆⋂{Vmi:i∈{1,2,…,n}}⊆⋂{↑mi:i∈{1,2,…,n}}⊆↑y$(⋆⋆⋆)

for some M00Mx. Finally, we show ⋂{↑b : bM00} ⊆ ▴𝓜y. For every x′ ∈ ⋂{↑b : bM00} and every M′ ∈ 𝓜 with sup M′ ⩾ x′, let $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{{M}^{\prime }}^{{x}^{\prime }}}\end{array}$ be the net defined in Remark 3.2 (3). Then $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{{M}^{\prime }}^{{x}^{\prime }}}\stackrel{\mathcal{M}}{\to }{x}^{\prime },\end{array}$ and thus $\begin{array}{}\left({x}_{\left(d,D\right)}{\right)}_{\left(d,D\right)\in {D}_{{M}^{\prime }}^{{x}^{\prime }}}\stackrel{\mathcal{T}}{\to }{x}^{\prime }.\end{array}$ It follows from Condition (⋆⋆⋆) and Remark 3.2 (4) that there exists $\begin{array}{}{M}_{0}^{{}^{\prime }}⊑{M}^{\prime }\end{array}$ such that

$⋂{↑a′:a′∈M0′}⊆⋂{Vmi:i∈{1,2,…,n}}⊆⋂{↑mi:i∈{1,2,…,n}}⊆↑y.$

This shows x′ ∈ ▴𝓜y, and thus ⋂{↑b : bM00} ⊆ ▴𝓜y. Therefore, P is an α*(𝓜)-continuous poset. □

Combining Lemma 3.15 and Lemma 3.16, we deduce the following result.

#### Theorem 3.17

Let (P, 𝓜) be a PM-space. The following statements are equivalent:

1. P is an α*(𝓜)-continuous poset.

2. For any net (xi)iI in $\begin{array}{}P,\left({x}_{i}{\right)}_{i\in I}\stackrel{\mathcal{M}}{\to }x\in P⟺\left({x}_{i}{\right)}_{i\in I}\stackrel{{\mathcal{O}}_{\mathcal{M}}\left(P\right)}{\to }x.\end{array}$

3. The lim-inf𝓜-convergence in P is topological.

#### Proof

(1) ⇒ (2): By Lemma 3.15.

(2) ⇒ (3): Clear.

(3) ⇒ (1): By Lemma 3.16.□

#### Corollary 3.18

([8]). Let (P, 𝓜) be a PM-space with 𝓢0(P) ⊆ 𝓜 ⊆ 𝓟(P). Suppose𝓜a ∈ 𝓜 and {yP : (∃ zP) yα(𝓜)z≪α(𝓜)a} ∈ 𝓜 holds for every aP. Then the lim-inf𝓜-convergence in P is topological if and only if P is α(𝓜)-continuous.

#### Proof

(⟹): To show the α(𝓜)-continuity of P, it suffices to prove sup▾𝓜a = a for every aP. Since the lim-inf𝓜-convergence in P is topological, by Theorem 3.17, P is an α*(𝓜)-continuous poset. This implies that there exists Ma ∈ 𝓜 such that sup Ma ⊆ ▾𝓜a and sup Ma = a for every aP. By Proposition 3.7, we have ▾𝓜a ⊆ ↓a. So sup ▾𝓜a = a.

(⟸): By Proposition 3.9 and Theorem 3.17.□

## Acknowledgement

This work is supported by the Doctoral Scientific Research Foundation of Hunan University of Arts and Science (Grant No.: E07017024), the Significant Research and Development Project of Hunan province (Grant No.: 2016JC2014) and the Natural Science Foundation of China (Grant No.: 11371130).

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## Footnotes

• 1

From the logical point of view, we stipulate ⋂{↑a ∩ ↓b : aA0 & bB0} = P if A0 = ∅ or B0 = ∅.

• 2

From the logical point of view, we stipulate ⋂{↑m : mM0} = P if M0 = ∅.

Accepted: 2018-07-17

Published Online: 2018-09-18

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1077–1090, ISSN (Online) 2391-5455,

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